# Order of reactivity of hydrogen halides

Solomon and Fryhle $$11^{th}$$ edition says :

Order of reactivity of HX in the acid catalyzed conversion of alcohols to alkyl halides follows the order : HI>HBr>HCl

We can account for this by saying that the acidity order is also HI>HBr>HCl and hence the $$-OH$$ is easily protonated by the more acidic halide but the reaction is already acid catalyzed by a strong acid and the conversion of $$-OH$$ to $$-H_2O^+$$ is fast so the acidity of $$HX$$ should make a negligible difference.

The slow step of the reaction is the formation of the carbocation and that does not seem to involve the halide so how can we explain this order ?